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Extreme Points of Vector Functions EXTREME POINTS OF VECTOR FUNCTIONS SAMUEL KARLIN Several previous investigations have appeared in the literature which discuss the nature of the set T in »-dimensional space spanned by the vectors ifsdui, ■ ■ ■ , fsdp.n) where 5 ranges over all measur- able sets. It was first shown by Liapounov that xi ui, • • ■ , un are atomless measures, then the set F is convex and closed. Extensions of this result were achieved by D. Blackwell [l] and Dvoretsky, Wald, and Wolfowitz [2]. A study of the extreme points in certain function spaces is made from which the theorems concerning the range of finite vector measures are deduced as special cases. However, the extreme point theorems developed here have independent interest. In addition, some results dealing with infinite direct products of func- tion spaces are presented. Many of these ideas have statistical in- terpretation and applications in terms of replacing randomized tests by nonrandomized tests. 1. Preliminaries. Let p. denote a finite measure defined on a Borel field of sets $ given on an abstract set X. Let Liu, 5) denote the space of all integrable functions with respect to u. Finally, let Miß, %) be the space of all essentially bounded measurable functions defined on X. It is well known that Af(/i, 5) constitutes the conjugate space of Liu, %) and consequently the unit sphere of Af(/x, %) is bi- compact in the weak * topology [3]. Let (®Ln) denote the direct product of L taken with itself » times and take i®Mn) as the direct product of M n times. With appropriate choice of norm, i®M") becomes the conjugate space to (®7,n). Another description of i®Mn) is that it consists of all «-vectors, each component of which is an essentially bounded measurable function. In notation, let xEi®Mn) be denoted by x=(x,(/)), i=l, ■ • -, », with x,- in M. 2. Extreme points in Ma. Let A be a bounded closed convex set in Euclidean »-space. Let Ma consist of all x in ( <8>Mn) whose range of values lie almost everywhere in A. I.e., for almost every t, x(i) = (x,(<)) is in A. Let B consist of the extreme points of A, and let B = closure of B. For n^3, it is not necessarily true that B = B. In view of the convexity of A, the set Ma is easily verified to be convex, bounded, and weak * closed. We indicate the proof of this last fact. Let x0 represent a weak * limit point of MA. Suppose xoEMa- This implies the existence of a set of positive finite measure Received by the editors October 11, 1952 and, in revised form, October 30, 1952. 603 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 604 SAMUEL KARLIN [August E with x0(E)(¡.A. Since a denumerable number of hyperplanes de- termine A, it is easy to construct a plane {£<} so that Zt-i ¡tiVi>c lor {r]i} Ç_A while 23"_i £¿t?(Ei) <c —e for a set of positive measure EiQE. Let w(t) denote the function in ®Ln(p, %) defined as fol- lows: ,, / ,. íiMEi) lor tin Eu \ w(t) = ( wat) = { > ). We note that (w-x)= ^2?_i JwiXidp represents the inner product and is èc for x in MA, while (wxa) <c —e. This shows that xQcannot be a weak * limit point of MA, a contradiction from which we conclude that XoÇ:MA- Since the unit sphere (® Mm) is bicompact in the weak * topology, we deduce that Ma is bicompact. On account of the bicompactness and convexity, the Krein-Milman theorem guarantees the existence of extreme points in Ma. The first theorem characterizes such ex- treme points. We define Mb in a similar manner to MA. Theorem 1. The extreme points of Ma are contained in Ms- Proof. Let xo = (Xf) be an extreme point of M Aand let us suppose the contrary that x0 is not in Mb. There exists an e0 such that xQ(£Ma(eô) where B(e0) consists of the closure of the set of points of A obtained from B by describing an e0 sphere about every point in B. In fact, otherwise, let en—>0 and suppose *oG-^B(í„> for every n. That is, except for a set En of p measure zero the range of values of x0 is in Mb^„). Since p( Z^") =0, we deduce that *0 is in Mr\B<.t^= Mb, which serves as a contradiction. Thus there exists a set E of positive meas- ure such that xo(t), lor t in E, is not in B(eo). It follows easily that a constant vector ä exists of small magnitude such that xo + â is in A when t is in E0(ZE. Put (â, l G Eo, $(t) = \ lo, t(£Eo. Consequently, Xo= (xo+<t>)/2-\-(x0 —$)/2 with x0 + #, x0 —$ in MA. This impossibility establishes the result, Q.E.D. Remark. The set Mb need not be contained in (MA)U) (extreme points) but Mb is. 3. Extreme points of Ma with side conditions. A finite measure p is said to be atomless if for any measurable set E of positive meas- ure p there exists a measurable subset Eo such that 0<p(Eo) <p(E). The theorem of Liapounov [4] states that if plt ■ ■ ■ , pn are atom- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 19531 EXTREME POINTS OF VECTORFUNCTIONS 605 less finite measures defined on the same Borel field of sets, then the span in Euclidean w space of the « tuples (/x(<)d/iy(i)) obtained by al- lowing x(£) to range over all characteristic functions is convex and closed. Therefore, it follows that the set of all » tuples ifxit)dp.jit)) with 0 =x(¿) ^ 1 spans the same set. An apparent weaker formulation of the above spanning result which is, however, equivalent is that for any given measurable set S there exists a set EES such that simultaneously UiiE)=miS)/2 for i=\, ■ ■ • , ». Let M [A, Uj] = [I,-,i= 1, • • • , »] with í,=/x(/)¿/tj(<) and x in Ma. It is of course understood that x is measurable with respect to all ßj. This set can also be viewed as points in Euclidean nm space. Theorem 2. 7/ju, (t'=l, ■ • • , ») consist of atomless finite measures, then the extreme points of the set T of all x(<) measurable /x,-(i = 1, • • •, w) satisfying xEMa and bi^fxdßi^äi are contained in Mb. Remark 1. The vector inequality fxdßi^äi means that the in- equality holds for each component of the vectors. Remark 2. The interest of Theorem 2 is that if we impose linear side conditions on the set Ma of the form 5¿^/xá/i¿^a¿, then no new extreme points are added to this convex subset of Ma when B = B. Only the set of extreme points of M a may be diminished. This result is in sharp contrast to the situation in the case of atomic meas- ures. We leave it to the reader to construct examples involving atomic measures which violate the conclusions of the theorem. Remark 3. If ß=ßX-\- ■ ■ • +/i„, then by virtue of the Radon- Nikodym theorem ßiiE) =fEfiit)dßit). The space Miß, %) considered here is the set of all bounded measurable functions x(¿), which is the conjugate space to 7,(/u, g) where % is the common Borel field of sets over which all the p¡ are defined. Therefore, bounded weak * closed sets in Miß, %) are bicompact and the same holds for (®M(ju, %)n). The statement of the theorem in terms of the /< is that the set Y of extreme points of all x(/) in MA which satisfy 5,¿/x/í(/)¿m(0 ^<*¿ is contained in Mb when ß is atomless. Since A is convex and closed and the linear restrictions are generated by elements of L iß, %), we deduce that T is bicompact and convex and hence possesses extreme points. The theorem thus determines the form of such extreme points. Proof of Theorem 2. Let x0 be an extreme point of T and suppose x0 is not in Mb. Then an argument as in Theorem 1 shows that for some positive eo the point x0 is not in ikfs(eo). Consequently there exists a set 5 of positive measure ß for which x0(¿) E-ß(«o) for t in 5. As in Theorem 1, one can find a constant vector c and a measurable set EES of positive ß measure such that x0±c, for t in E, is contained License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 606 SAMUELKARLIN [August in A. An application of the theorem of Liapounov yields a measur- able subset Eo of E with Pi(Eo)= I fidp = — I fidp = — pi(E) for i = 1, , n. J B<¡ 2 J B 2 Put c, t in Eo, — c, t in £0, fit) ■c, t in E — Eo, and #(i) = c, tinE— Eo, 0 elsewhere, 0 elsewhere. It is easy to verify that x0= (£o+<?)/2 + (x0+!r')/2 where *o+# and ¿0+$ are in Af¿. A simple computation gives that /(*o + $)f>dp= I xofidp+ c\ I /¿¿/i - I /id/t «J LVi:0 J E-E0 J = I *o/¡¿M+ C — I /.¿M— — I /¿¿Ai = I *o/i¿M- Similarly, f(xo-r-$)fidp = fxofidp. Therefore ío+í and *o+^ satisfy the linear inequalities and hence lie in T. We have exhibited a con- tradiction of the extreme point nature of x0 and the proof of the theorem is hereby complete.
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